Problem: The pressure at sea level is $1$ atmosphere and increases at a constant rate as depth increases. When Sydney dives to a depth of $23$ meters, the pressure around her is $3.3$ atmospheres. The pressure $p$ in atmospheres is a function of $x$, the depth in meters. Write the function's formula. $p=$
Answer: The pressure increases at a constant rate as depth increases, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $p= mx+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the pressure at sea level is $1$ atmosphere, so the $y$ -intercept ${b}$ is ${1}$, and our function looks like $p={m}x+{1}$. We also know that at a depth of $23$ meters, the pressure is $3.3$ atmospheres, which means when $x=23$, $p=3.3$. We can use this and the $y$ -intercept to find ${m}$ : $\begin{aligned} {m}&=\dfrac{p_2-p_1}{x_2-x_1} \\\\ &=\dfrac{3.3-1}{23-0} \\\\ &=\dfrac{2.3}{23} \\\\ &={0.1} \end{aligned}$ This means pressure increases at a rate of $0.1$ atmospheres per meter. Since ${m}={0.1}$ and ${b}={1}$, the desired formula is: $p={0.1} x + {1}$